Cube, under stress

Andrade creep n. A type of creep behavior in which the comphance of the sample is proportional to the cube root of the time under stress. A variety of polymers exhibit this behavior. 

Fig. 1.10 An elementary cube, removed from a structural material, is shown under stress-like forces. These are the components of a stress tensor. Often, the coordinate system is shown as indicated on the left of this figure...

T is often referred to it as traction and clearly it represents a stress vector acting parallel to the surface. From the Pythagorean relation shown in Fig. 1.12, the sum of the squares of the normal and tangential stresses on any face of an elementary cube under arbitrary stress is equal to their sum. Furthermore, the shear component, T, is usually not parallel to any of the axes of a chosen coordinate system, as indicated in Fig. 1.12. It is, however, common to resolve this shear stress into two components, each of which is parallel to a chosen reference coordinate system. In Fig. 1.13, the stresses are indicated on the z plane. 

Deformation of elastic cube under shear stress S. ... 

Figure 10.1. The forces on the faces of a unit cube in a body under homogeneous stress.

The simplest state of stress is one in which the fluid element in the cube is under hydrostatic pressure only. In that case, T — —where 8 is the unit tensor ... 

The SI units are N Grouping nine components together into a 3 x 3 matrix does not automaticcdly lead to a tensor for that other criteria should hold, such as invariance under change of coordinates. In 13.6.1) the three diagonal terms (r, T, T ), known as normal stresses, act normal to the six faces of a cube of the... 

Figure 5.8 illustrates what can happen when an amount of material is put under a given stress for some time. Envisage, for instance, a cheese cube onto which a weight is placed and after some time removed. In (a) the stress-time relation applied is shown, and in (b) we see the response of a... 

Also, life can t make a cube out of carbon. Carbon s four bonds form 109-degree angles naturally, not the 90-degree angles required for square shapes. Under duress, a microbe can squeeze carbon into a square, but a cube has too many stressed angles. Carbon cubes are explosive, which is in general not a good property for life. Cubes will be made by life, but out of iron and sulfur, not carbon. 

Consider now a generic surface of an anisotropic three-dimensional objeet (Fig. 11.5(c)), and a stress vector at decomposed along three directions, the one perpendicular to the surface corresponding to direct stress, and the other two forming what is called shear (tangential) stress. Considering an elementary cube within the material, there is one such stress vector with its three components for each face, so there are nine stress components, only six of which are independent because of the condition cry = oji in mechanical equilibrium (no net displacement under shearing stress). The shear modulus G is defined as... 

This approximate calculation gives values for Og about one thousand times greater than those observed. For example, sodium chloride fractures on its cube plane under a stress of 2.2 MPa, whereas the value of jg is 5 GPa. 

A usual metal crystal contains about 10 dislocation lines/cm. Moving all of these out of the crystal under the influence of an applied stress does not account for the observed shear strain. Suppose our crystal is a cube 1 cm on a side and it is oriented in such a way that a shear stress is applied on the slip plane in a possible slip direction. Suppose one-third of the 10 dislocation lines have this slip direction and are oriented favorably for slip (this is clearly an overestimate), then taking b equal to 3 x 10 cm gives a total shear strain of 1 %. Shear strains much greater than this are observed, sometimes exceeding 100% in single crystals. Clearly, dislocation multiplication processes... 

A cube of the epoxy of Problem 12.4 and having unit volume is put under a tensile stress of 10 MPa. What is the percent change in volume ... 

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